6

Controllable
Decomposition

qLet be a state space realization of a
(not necessarily stable) transfer matrix *G*(*s*). Suppose that there exists a symmetric matrix

(*P* is not necessarily the
Gramian) with *P**1* nonsingular such that

Now partition the
realization (*A,B,C,D*) compatibly with *P*
as

Then is also a
realization of *G.*

Moreover, (*A**11**,B**1*) is controllable if *A**11* is
stable.